Analysis of discontinuous Galerkin schemes for flow and transport problems
| dc.contributor.advisor | Riviere, Beatrice | en_US |
| dc.creator | Masri, Rami | en_US |
| dc.date.accessioned | 2022-09-26T16:38:45Z | en_US |
| dc.date.available | 2022-11-01T05:01:14Z | en_US |
| dc.date.created | 2022-05 | en_US |
| dc.date.issued | 2022-04-21 | en_US |
| dc.date.submitted | May 2022 | en_US |
| dc.date.updated | 2022-09-26T16:38:45Z | en_US |
| dc.description.abstract | We formulate and theoretically analyze interior penalty discontinuous Galerkin (dG) methods for flow and transport problems. In particular, the analyses of dG formulations for (1) non-linear convection diffusion equations, (2) incompressible Navier– Stokes equations, (3) Cahn–Hilliard–Navier–Stokes equations, and (4) elliptic and parabolic problems with a Dirac line source are presented. First, we formulate a new locally implicit dG method for nonlinear convection diffusion equations, show that this scheme yields a less restrictive constraint on the time step, and prove optimal error estimates. This formulation is motivated by applications to coupled systems of solute transport and blood flow where it is combined with a Runge–Kutta dG scheme to simulate these systems in one dimensional vessel networks. The second scheme we analyze is a pressure correction dG scheme for the incompressible Navier–Stokes equations in two and three dimensional domains. Studying this scheme is motivated by its ability to efficiently simulate flow in large-scale complex computational domains. We show unconditional stability, unique solvability, and convergence of the discrete velocity by obtaining error estimates. The derivation of these error estimates requires the development of several tools including new lifting operators. Further, optimal error estimates in the L2 norm for velocity are obtained via introducing dual Stokes problems. To complete this analysis, we also show convergence of the discrete pressure. The pressure correction dG approach is extended to the Cahn–Hilliard–Navier– Stokes equations within a decoupled splitting framework. We show that the proposed scheme is uniquely solvable. The discrete mass conservation, the energy dissipation, and the L∞ stability of the order parameter, are established. We prove optimal a priori error estimates in the broken gradient norm. Using multiple duality arguments, we obtain an optimal error estimate in the L2 norm. The stability proofs and error analysis are based on induction arguments without any regularization of the potential function. The third class of problems we consider are elliptic and parabolic problems with a Dirac line source. Such problems are used to couple one dimensional flow models in blood vessels to three dimensional models in tissues. The analysis of such problems is challenging since the gradient of the true solution is singular. We propose dG discretizations of such problems and prove convergence in the global L 2 norm. For the elliptic problem, we show convergence in weighted energy norms. In addition, we show almost optimal local error estimates in the L 2 and energy norms in domains excluding the line. For the parabolic problem, we establish global error estimates for the semidiscrete formulation and for the fully discrete backward Euler dG discretization. | en_US |
| dc.embargo.terms | 2022-11-01 | en_US |
| dc.format.mimetype | application/pdf | en_US |
| dc.identifier.citation | Masri, Rami. "Analysis of discontinuous Galerkin schemes for flow and transport problems." (2022) Diss., Rice University. <a href="https://hdl.handle.net/1911/113379">https://hdl.handle.net/1911/113379</a>. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1911/113379 | en_US |
| dc.language.iso | eng | en_US |
| dc.rights | Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder. | en_US |
| dc.subject | Discontinuous Galerkin methods | en_US |
| dc.subject | Numerical analysis | en_US |
| dc.subject | non-linear convection diffusion equations | en_US |
| dc.subject | incompressible Navier– Stokes equations | en_US |
| dc.subject | Cahn–Hilliard–Navier–Stokes equations | en_US |
| dc.subject | Dirac line sources | en_US |
| dc.title | Analysis of discontinuous Galerkin schemes for flow and transport problems | en_US |
| dc.type | Thesis | en_US |
| dc.type.material | Text | en_US |
| thesis.degree.department | Computational and Applied Mathematics | en_US |
| thesis.degree.discipline | Engineering | en_US |
| thesis.degree.grantor | Rice University | en_US |
| thesis.degree.level | Doctoral | en_US |
| thesis.degree.name | Doctor of Philosophy | en_US |
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